Closed superstring moduli tree-level two-point scattering amplitudes in Type IIB orientifold on $T^6/(Z_2\times Z_2)$

We reconsider the two-point string scattering amplitudes of massless Neveu-Schwarz--Neveu-Schwarz states of Type IIB orientifold superstring theory on the disk and projective plane in ten dimensions and analyse its $\alpha'$ expansion. We also discuss the unoriented Type IIB theory on $T^6/\mathbb{Z}_2\times\mathbb{Z}_2$ where two-point string scattering amplitudes of the complex K\"ahler moduli and complex structures of the untwisted sector are computed on the disk and projective plane. New results are obtained together with known ones. Finally, we compare string scattering amplitudes results at $\alpha'^2$-order with the (curvature)$^2$ terms in the low energy effective action of D-branes and $\Omega$-planes in both cases.